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We obtain scalar hairy black holes from Einstein-Maxwell-conformally coupled scalar (EMCS) theory with the scalar coupling parameter α to the Maxwell term. In case of α=0, the α=0 EMCS theory provides constant (charged) scalar hairy black hole and charged BBMB (Bocharova-Bronnikov-Melnikov-Bekenstein) black hole where the former is stable against full perturbations, while the latter remains unstable because it belongs to an extremal black hole. It is noted that for α≠0, the unstable Reissner-Nordström black holes without scalar hair imply infinite branches of n=0(α≥8.019),1(α≥40.84),2(α≥99.89),⋯ scalarized charged black holes. In addition, for α>0, we develop a single branch of scalarized charged black hole solutions inspired by the constant scalar hairy black hole. Finally, we obtain the numerical charged BBMB black hole solution from the α=0 EMCS theory.